In Derivatives Explained, we gave you an overview of the ways in which we
could make money by trading the cash instrument around an options position.
This article will introduce one specific set of techniques for describing the
behavioural characteristics of an options position or a portfolio of options,
futures, forwards and cash.
When we purchase an option, we can trade the cash instrument
(called "trading spot" or "trading the cash"), hoping to
realize more profit from trading the cash than we pay initially in premium for
the option. When we sell an option, we hope that the premium that we are paid
upfront dwarfs the losses we will sustain from trading the cash.
When we buy options, we are said to be buying volatility. We
make money if the spot rate is volatile enough for us to pay for the option.
When we sell options, we are selling volatility. We make money if spot is calm
enough that we don't have to hedge the exposure frequently.
However, delta hedging is not the only way for us to make money
with options. The genius of derivatives is that it allows us to take positions
in (or to hedge against fluctuations in) other aspects of the cash instrument's
price evolution. Derivatives are dangerous if we do not understand or address
each potential dimension of their risk.
Here are several examples. With a simple plain vanilla option,
we can make money if implied volatility moves in our favor. With currency
futures, currency forwards and currency options, we can speculate on the spread
between interest rates in two different countries for a maturity date. With
some exotic options, we can buy an option that appreciates in value with the
passage of time (all other things being constant) and that also appreciates in
value with movement lower in implied volatility.
Options dealers and savvy options traders use time-proven
techniques to break down the risks in an options position or in a portfolio of
options, futures, forwards and cash positions into information that is more
readily comprehensible and therefore more easily positioned or hedged. This
method of analysis employs tools called the "greeks", as well as
using simulation, scenario analysis and value-at-risk analysis.
We will discuss simulation, scenario analysis and value-at-risk
analysis in subsequent articles.
The greeks get their name from the fact that the sensitivities
of an option to various market parametres are labelled with letters from the
greek alphabet.
DELTA
The delta of an option is the sensitivity of the option's price
to very small changes in the price of the underlying instrument.
When we talked about trading spot around the options position
in order to realize profit that would pay for the option's premium, we were
talking about trading the delta.
By taking an opposite position equal in size to the option's
delta, we immunize the option against profit and loss variability due to small
changes in the spot rate.
For example, consider our equity call option with a strike price
of $50 when the underlying price is $50. Because it is an at-the-money option,
we know that the delta is 50. The delta is expressed in terms of a percentage
of the notional amount. An option that is hopelessly out-of-the-money very near
to expiration has a delta of 0. Also, near expiration, an option that is
completely in-the-money with no danger of being thrown out-of-the-money has a
delta of 100. Everything else is in between. At-the-money options have a delta
of 50.
Our equity call has a positive delta because it is a long
position in a call. If we exercise the call, we will end up being long the
stock.
An equity put struck at-the-money would have a negative delta
of 50. If we exercise the put, we will end up being short the stock.
Similarly, shorting a call implies a negative delta and
shorting a put implies a positive delta.
To delta hedge our long at-the-money equity call struck at $50,
we need to know the notional amount. Let it be $100 for the sake of argument.
Therefore, the delta position implied by our option is $50 (i.e. 50/100 x
$100).
If we take a short position in the cash market (assuming that
shorting the stock is feasible and liquid enough) at the spot price of $50, we
have immunized the option's sensitivity to small changes in the spot price.
If spot goes to $48, the $2 we make on the short stock position
will offset the $2 we will lose on the change in price of the option.
Similarly, if spot goes to $52, the $2 we make on the option premium will be
offset by the $2 we will lose on the short stock position.
Assuming that we own the option, if we plot the curve of the
option premium (on the y-axis) against the price of the underlying instrument
(on the x-axis), everything else remaining constant, we obtain a convex curve.
The slope of this convex curve is the option's delta.
GAMMA
Things begin to get interesting for larger moves in the stock
price.
If spot goes to $70, we might expect to make $1150 on the
option price while only losing $1000 on the short stock position.
How does this work?
Because of the convexity of the option's curve, the delta will
change if spot moves enough.
If spot goes to $52, the delta might change to 52. If spot goes
to $55, the delta might change to 57. If spot goes to $60, the delta might
change to 64. If spot goes to $70, the delta might be 80. The option position
behaves as if it is a miraculous trade that seemingly gets longer as spot goes
higher in a non-linear fashion.
Since we have only hedged our exposure to a position that is
long $50 at $50, the hedged option position will continue to make money on the
incremental position, i.e. the part that appeared to get longer from $50 to $70
at an average rate of say $65.
The greater the convexity of the option curve, the more bang
for our long option buck and the more pain we will endure if we are short the
option, in a volatile environment.
Convexity is described by the greek letter called
"gamma".
Mathematically, gamma is the second derivative of the option's
price with respect to the underlying cash price. Intuitively, it is the
sensitivity of the delta (or rate of change of the delta) with respect to the
cash price.
VEGA
We know that options will be expensive when volatility is
actually high or when volatility is thought to be heading higher. We also know
that options are cheap when volatility is low or when volatility is believed to
be heading lower.
There are two kinds of volatility between which we must
distinguish: actual volatility and implied volatility.
Actual volatility is a measure of
how much the spot price moves around, in fact, for a given time period.
Implied volatility is the
volatility used in the calculation of the option's price. Without going into
the mathematics of it at this point, suffice it to say that we can back out (or
"imply") the volatility used to calculate an option's price, if we
know with certainty the value of each of the other variables used in the option
valuation formula. For the Black-Scholes-Merton model, the list of these
remaining variables typically includes the underlying cash price, the maturity
date, the delivery date, the strike price and the risk-free rate of interest.
Some of the more developed derivatives markets, such as the
foreign exchange options market, actually trade in terms of implied volatility
or "vol" instead of specifying a price at which may buy or sell the
option in question.
The sensitivity of an option's price to changes in its implied
volatility, all other things being constant, is called the "vega".
Let us consider the case where we have just bought and delta
hedged the long $50 equity call in the stock of company ABC Inc. Spot does not
move for a couple of hours until a headline tells the market that DEF Inc. has
made a hostile bid for ABC Inc.
Even though spot does not move immediately because traders are
confused about the implications of the DEF bid, implied volatilities jump much
higher because of the additional uncertainty for ABC's future prospects posed
by the DEF initiative.
We will make money, not from delta hedging, but from the jump
up in the value of the option. We now own something that has become more
valuable in the blink of an eye because the market believes that the volatility
of the ABC stock price will be greater than previously thought. Pity the poor
short option holder.
OTHER GREEKS
There are other greeks that we will leave for future
discussion. For example, "rho" describes the option's sensitivity to
changes in the domestic interest rate.
In the next article, we will describe some of the ways in which
understanding the greeks can get tricky. Apparently, simple positions can look
very complex very quickly.
Article by Chand Sooran, Principal Victory
Risk Management Consulting, Inc. |